Why can't I type simple formulas when using transforms?
Activate formula mode for the Numeric input

That's because currently the "formula" input (advanced mode) is not active by default. This was made to preserve some old functionalities, like the / shortcut that, as in your case, transforms the number to its inverse.

The rule is that the formula mode becomes active only once the Numpad * (or the =) key has been pressed, so in your case you can type

Numpad *360/5

Or

3Numpad *60/5

36Numpad *0/5

360Numpad */5

and all should work as expected.



Here you can find the Sneak Peek video made by Plasmasolutions about the topic.

And here's some lines from the 2.70 release notes abot the UI (see Transform Tools Number Input section):


  Number input for tools (when you grab an object for example, you can type indirectly values with the keyboard) has been reworked, you can now use units and Python expressions in it. (87cc890aef53).
  
  We have two "modes", simple and advanced ones, use '=' or 'pad*' to
  enable advanced mode, and 'ctrl-=' or 'ctrl-pad*' to switch back to
  simple mode.
  
  Simple mode works nearly like it did before, it only accepts simple
  numbers, and you can still use '-' to negate and '/' to inverse the
  value (as well as non-number input shortcuts like XYZ, RSG, etc.).
  Compared to Blender 2.69 behavior, you can now copy and paste
  expressions, and navigate in the number you typed, so you can for
  example fix a typo without having to type everything again.
  
  In advanced mode, we gain more power and flexibility, but lose a few
  "shortcuts" like '-' to negate, or '/' to inverse




The 3282.81° issue can be explained with the fact that Blender makes some conversions between radiants and degree that, in conbination with the inverse operation, led to an apparently unexpected results.

By default the input for rotation are considered as degree (1° in the depicted case). In randiants would be:

rad(1°) = 0,0174532925199433


When you press / Blender uderstand that it has to caluculate the inverse of your input, so it switchs to radiants to perform the operations. The resulting input now  is

1/0,0174532925199433 = 57,2957795130823


And only at this point he perform the degree conversion:

deg(57,2957795130823) = 3282,80635001174°


To avoid this kind of I would suggest to use the advanced mode, where you get expected results:

* 1/1